Regular Subgraphs of Almost Regular Graphs - Semantic Scholar
JOURNAL
OF COMBINATORIAL
Regular
THEORY,
Series B 37, 79-91 (1984)
Subgraphs
of Almost N.
Department
Graphs
ALON*
of Mathematics, Massachusetts Institute Cambridge, Massachusetts
S. Institute
Regular
of Technology,
FRIEDLAND
of Mathematics, Hebrew University Jerusalem, Israel
of Jerusalem,
AND
G. Department
KALAI*
of Mathematics, Massachusetts Institute Cambridge, Massachusetts Communicated
by the Managing
of Technology,
Editors
Received July 25, 1983
Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k > 2q - 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q. r = q (mod 2)). It is also proved that every simple graph with maximal degree A > 2q - 2 and average degree d > ((2q - 2)/(2q - l))(A + 1), where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r = q (mod 2)). These results follow from Chevalley’s and Olson’s theorems on congruences. 0 1984 Academic Press, Inc.
1.
INTRODUCTION
In this paper we use the theorems of Chevalley [5] and Olson [9, lo] (and some extensions) on congruences, to prove the existence of regular subgraphs of certain graphs. All graphs considered are finite, undirected, and contain no loops, unless otherwise stated. Note that we allow multiple edges. A simple graph is a graph without multiple edges. * The contribution of these authors was supported in part by the Weizmann Fellowship for Scientific Research.
79 00958956/84 582b/37/1-6
$3.00
Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.
80
ALON,
FRIEDLAND,
AND
KALAI
A graph H is q-divisible if q divides the degree of every vertex of H. Detinef(n, q) to be the maximal number of edges of a graph G on n vertices, that contains no nonempty q-divisible subgraph. In Section 3 we prove that
provided q is an odd prime power. If q is a power of 2, then
We further show that in both inequalities equality holds for all n > 3 and that a graph on n > 3 vertices and e >f(n, q) edges contains at least 2e-fh7) - 1 nonempty q-divisible subgraphs. Note that for q = 2 this is just the well-known fact that the dimension of the cycle space of G is at least e-n+ 1. For k < s a graph G is of type (k, s) if the degree d(v) of every vertex of it satisfies k < d(v) < s and G is not k-regular. In Section 4 we show that if q is a prime power, q > r, q = r (mod 2) and k > 2q - 2 then every graph G of type (k, k + 1) contains an r-regular subgraph. In particular: Every 4-regular graph plus one edge contains a 3-regular subgraph. (1.1) This result is closely related to a well-known conjecture of Berge and Sauer (see, e.g., [4, p. 2461) that asserts that every 4regular simple graph has a 3-regular subgraph. Some positive results about this conjecture can be found in [6], and in [ 111 Taikinov announced that he verified it. However, the Berge-Sauer conjecture is false for graphs with parallel edges; (every graph obtained from an odd cycle by replacing every edge by two parallel edges forms a counterexample). Therefore, the “plus one edge” cannot be omitted in (1.1). A short derivation of (I. 1) from Chevalley’s theorem appears separately in [ 11. In Section 4 we also show that if q is a prime-power, q > r and q = r (mod 2) then every simple graph G with maximal degree d > 2q - 2 and average degree d > ((29 - 2)/(2q - l))(d + 1) contains an r-regular subgraph. In particular, every simple graph with maximal degree d > 4 and average degree d > $(d + 1) contains a 3-regular subgraph. This result may help in solving a long standing problem of Erdiis and Sauer (see, e.g., [2, p. 399, problem 201). They asked for an estimation of the maximal number of edges of a simple graph on n vertices that contains no 3-regular subgraph. Our paper is organized as follows. In Section 2 we describe the algebraic tools: we quote the theorem of Olson, show how it is related to the classical theorem of Chevalley, and obtain a simple corollary. In the Appendix we prove an extension of Olson’s theorem and apply it to graph theory. Our proof is different from Olson’s proof and is somewhat similar to the proof of
81
REGULAR SUBGRAPHS
Chevalley’s theorem given in [5]. In Section 3 we derive the results on q-divisible subgraphs and in Section 4 we combine these with known results of Petersen, TaOkinov, Thomassen, Tutte, and Vizing to obtain our results on regular subgraphs.
2. THE ALGEBRAIC
TOOLS
Our main algebraic tool in this paper is the following result of Olson [9]. Let p be a prime and suppose d, > d, > . . . > THEOREM 2.1 (Olson). d, > 0. For 1 < i < m let a”’ = (a?‘, a:“,..., a’,“) be a vector with integer coordinates. If m > k (pdj-
l)
j=l
then there exists a subset 0 # Z c ( 1,2,..., m) such that
c {a;“: i E I} E 0 (modpdi),
j = l,..., n.
(2.1)
It is worth noting that for d, = d, = ..a = d, = 1 it is possible to derive this result from the classical theorem of Chevalley (see, e.g., [5]). Indeed consider the following system of polynomial equations 1$, aji’xf-’
= 0 (modp),
j=
I,..., n.
Clearly xi = 0 is a solution. Since the left-hand side of each equation is of degree p - 1 at most, the Chevalley’s theorem ensures a nontrivial solution if m > (p - 1) it. As xp-’ = 1 (mod p) for x f: 0 (modp), the existence of a nontrivial solution implies the assertion of Theorem 2.1 in this case. In the Appendix we prove a generalization of Olson’s theorem. Our proof is different from Olson’s proof and is somewhat similar to the proof of Chevalley’s theorem given in [5]. COROLLARY
2.2.
Suppose d, 2 d, > ... > d, > 1 and let p and a”’
(1 < i( m) be as in Theorem 2.1. If i j=l
a;!) G 0 (modp)
for
i = 1, 2,..., m
82
ALON,
FRIEDLAND,
AND
KALAI
and n-1 m
>p4-I
-
l+
x
(pdj-1)
j=l
then the conclusion of Theorem 2.1 holds. ProoJ For every 1 < i < m, define a vector b”’ = (bli),..., bc’) as follows: bji) = asi’ for 1 <j < n - 1 and bt’ = l/p CJ= I ali’. Applying Theorem 2.1 to the vectors b”), we conclude that there exists a subset 0 # Z c ( l,..., m) such that C {aj”: i E I} = c
{bji): i E I) s 0 (modpdj)
for 1 <j d, > ea. > d, , we conclude i E I} E 0 (modpdj) for all 1 <j < n. B
It is worth noting that both Theorem 2.1 and Corollary 2.2 are best possible. Indeed, let e”’ be the standard vector (Sj, ,..., Sj,), j = l,..., It. A set of pdj - 1 copies of e”’ for 1 <j < n shows that Theorem 2.1 is best possible. A set of pdj - 1 copies of e(j) - e’“’ for 1 Q j < n - 1 plus p d,-1 - 1 copies of pe W) shows that Corollary 2.2 is best possible. Theorem 2.1 and Corollary 2.2 can be used to prove the existence of one 0 # z c { 1, 2,..., m} that satisfies (2.1). Combining them with the following proposition of Olson [lo] we conclude that if m is large enough there are many such Is. PROPOSITION 2.3 (Olson). Let H be an abelian group and suppose that for every h I ,..., h,, , E H there exists a subset 0 # Z c { l,..., g + 1) such that
c (hi: i E I} = 0.
(2.2)
Zfh , ,..., h,,, E H and I > 1, then there exist at least 2’ - 1 distinct subsets Z, 0fzc {l,..., g + I} that satisfy (2.2). 3.
q-DIVISIBLE
SUBGRAPHS
Our main task in this section is to estimate the function f(n, q) defined in Section 1. Recall that f(n, q) is the maximum number of edges of a graph G
83
REGULARSUBGRAPHS
on n vertices that contains no nontrivial q-divisible subgraph. Clearly f(n, 1) = 0 and f(n, 2) = n - 1. The following theorem is an easy consequence of Theorem 2.1 and Corollary 2.2. THEOREM
3.1.
ft%4) 3. Define g(n,k)=(k-
1)-n
=(k-l)*n-(k/2)
if
k is odd,
if
k is even.
For an odd integer k > 1 let G,(k) denote the Shannon triangle obtained from a triangle by replacing each edge by k - 1 parallel edges. Similarly, for even k, let G,(k) be the graph obtained from a triangle by replacing two edges by k - 1 parallel edges each and the third edge by (k/2) - 1 parallel edges. For n > 3 let G, = (V,, E,J be a graph obtained from G,(k) by adding to it n - 3 new vertices and joining each by k - 1 edges to vertices of G,. Clearly 1V,l = n and JE,I = g(n, k). One can easily check that G, contains no nontrivial k-divisible subgraph. Combining this with Theorem 3.1 we obtain PROPOSITION
(ii)
3.2.
(i)
For n > 3 and every k, f(n, k) > g(n, k).
If q is a prime power then f(n, q) = g(n, q).
There is some interest in considering separately the case of simple graphs. Thus we definef,(n, q) as the maximal number of edges of a simple graph G on n vertices that contains no nontrivial q-divisible subgraphs. Clearly f,(n, q) q* - 1 equality holds.
84
ALON,
PROPOSITION 3.3.
(ii)
FRIEDLAND,
AND
KALAI
(i)
For every odd integer k > 1 and every n > k2 - 1
f,h
k) > g(n, k) = (k - 1) . n.
If q is an odd prime power
and n > q2 - 1 then
f,(n,q)=g(n,q)=(q-l).n. Proof. Part (ii) follows immediately from (i) and Theorem 3.1. To prove part (i) we construct a suitable example. Let tG denote the disjoint union of t copies of the graph G. Let G + H denote the join of the graphs G and H, i.e., the graph obtained from their disjoint union by joining each vertex of G to each vertex of H. Let Ek-, be the graph consisting of k - 1 isolated vertices, let K,,,- , denote the star with k - 1 edges and define G,,=JIZ~-~ + (k-
l)Kl,+l.
Suppose n > k2 - 1. Let G = (V, E) be a graph obtained from G, by adding to it n - (k2 - 1) new vertices and joining each of them to k - 1 vertices of G,. One can easily check that 1VI = n and 1E I= (k - 1) . n. In order to complete the proof we must show that G contains no nontrivial k-divisible subgraph. Clearly it is enough to show that G, contains no such graph. G, has vertices of three different types: let us call these of degree k(k - 1) vertices of type 1, these of degree 2k - 2- of type 2, and these of degree k- of type 3. Suppose G, has a nontrivial k-divisible subgraph H = (Y’, E’), where dH(v) > 0 for all v E V’. We claim that V’ contains all k - 1 vertices of type 1. Indeed, otherwise V’ contains no vertex of type 3 (since its degree in H is 1, then G contains at least 2’ - 1 nontrivial q-divisible subgraphs. 1
Note that since a 2-divisible
subgraph is just an Eulerian-subgraph,
for
q = 2 the last theorem is the well-known fact that the dimension of the cycle space of G is at least e - n + 1. Remark 3.6. Let G = (V, E) be a directed graph. For e E E and v E V put a:‘= + 1 (-1) if e goes out of (into) v and a:,@= 0 otherwise. If V = {V , ,***,v,,} define a’@ = (at’,..., a:). Since ,JJj”=, aI;’ = 0, one can apply Theorem 2.1 to the vectors (a!:‘,..., a~~‘-,) and show that if q is a prime power (even or odd) and JE1> (q - 1) e (n - 1), then G contains a subgraph H such that q 1d,+(v) - d;(v) for all u E V. This easily implies that every bipartite graph G with n vertices and more than (q - l)(n - 1) edges contains a nontrivial q-divisible subgraph.
We close this section with a conjecture. Conjecture 3.1.
For every n > 3 and every k f(n, k) < (k - 1) . n.
86
ALON,FRIEDLAND,AND
4. REGULAR
Recall the definition prove the following.
SUBGRAPHS OF ALMOST
KALAI REGULAR
GRAPHS
of a graph of type (k, s) given in Section 1. Here we
THEOREM 4.1. Let G be a graph of type (k, k + 1) and let r be an integer, If q is a prime power, q > r, q = r (mod 2), and k > 2q - 2 then G contains an r-regular subgraph. THEOREM 4.2. Let G be a graph of type (k, k + 2) and let r be an integer. If q is a prime power, q > r, q E r (mod 2), and k > 2q - 1 then G contains an r-regular subgraph.
Note that by Bertrand’s postulate (for every r there is a prime between r and 2r), Theorem 4.2 implies that if k > 4r then every graph of type (k, k + 2) contains an r-regular subgraph. In fact, the known improvements of Bertrand’s postulate (see, e.g., [2, p. xx]) enable one to show that Theorem 4.2 implies that for every E > 0 if r > r(s) is odd and k > (2 + E) r then every graph of type (k, k + 2) contains an r-regular subgraph. Shannon’s triangle obtained by replacing each edge of a triangle by r - 1 parallel edges shows that this is close to being best possible. For simple graphs we prove THEOREM 4.3. If q is aprimepower, q > r, and q s r (mod 2) then every simple graph G with maximal degree A > 2q - 2 and average degree d > ((29 - 2)/(2q - l))(A + 1) contains an r regular subgraph.
These theorems are proved by combining Theorem 3.1 with results of Petersen, TaBkinov, Thomassen, Tutte, and Vizing. In what follows we state these results and prove our theorems. LEMMA 4.4 (Thomassen [ 121, a somewhat weaker version was proved by Tutte [ 131.). Let G be a graph of type (k, k + 1) and suppose0 < r < k. Then G contains a spanning subgraph of type (r, r + 1). LEMMA 4.5 (Proved by Petersen [4, p. 751 for even k, and by Taikinov [ 111 for odd k.). If k > r, k = r (mod 2), then every k regular graph contains an r regular subgraph.
Proof of Theorem 4.1. By Lemma 4.4, G contains a (spanning) subgraph L=(V,E) of type (2q-2,29-l). Clearly ]E]>f.]v] (2q-2)= (q- 1). IV]. Th eref ore, by Theorem 3.1, L contains a nontrivial q-divisible subgraph H. However, for every v E V deg,(v) Q deg,(v) < 2q - 1 and thus H is q regular. The result now follows from Lemma 4.5. 1
REGULARSUBGRAPHS
87
For the proof of Theorem 4.2 we need LEMMA 4.6. Let G be a graph of type (k, k + 2) and suppose0 < I < k - 2. Then G contains a spanning subgraph of type (r, r + 2) with average degree strictly greater than r + 1.
Outline of Proof The proof is very similar to the proof of Thomassen [ 121 to Lemma 4.4. His argument easily shows that G contains a spanning subgraph L of type (r + 2, r + 4). The same argument shows that L contains a spanning subgraph H of type (r + 1, r t 3) with at least one vertex of degree r + 2, and that H has a spanning subgraph of type (r, r t 2) with more vertices of degree r t 2 than vertices of degree r. We omit the details. 1 Proof of Theorem 4.2. By Lemma 4.6, G contains a (spanning) subgraph L=(V,E)oftype(2q-3,2q-l)withIEl>(q-l)1VI.Thissubgraph,as is shown in the proof of Theorem 4.1, contains an r-regular subgraph. a
For the proof of Theorem 4.3 we need the following well-known result of Vizing (see, e.g., (2, pp. 230-2321). LEMMA 4.7 (Vizing). The edges of every simple graph with maximal degree A can be covered by A t 1 disjoint matchings.
Proof of Theorem 4.3. $ y L emma 4.7 the edges of G = (V, E) can be covered by A t 1 disjoint matchings. Let L = (V, E’) be the graph on V consisting of the edges of the 2q - 1 biggest matchings. Clearly
IE’l>~~E~=~.+d.lP’ (2q-1)(2q-2)(A+1)~Y~=(q-1)/YI. >2(A t 1) (2q - 1)
By Theorem 3.1, L contains a nontrivial q-divisible subgraph H, which is, as in the proof of Theorem 4.1, q-regular. The result follows from Lemma 4.5. I Remark 4.8. (a) We can slightly improve the constant (2q - l)/ (2q - 2) in Theorem 4.3 but this makes the proof somewhat more com-
plicated. (b) Remark 3.6 and Kiinig’s theorem (see, e.g., [4, p. 93, Theorem 6.11) enables us to obtain the following improvement of Theorem 4.3 for bipartite graphs: If q is a prime power, q > r then every bipartite graph G with maximal degree A and average degree d > ((2q - 2)/ (2q - 1)) A contains an r-regular subgraph.
88
ALON,FRIEDLAND,AND
KALAI
(c) Combining the result of [8] with Theorem 4.2 one can easily prove the following: Let G be a graph of type (k, k + s) and let r be an integer. If q is a prime power, q > r, q- r (mod 2), k> 2q - 1, and s/k < 2/(2q - l), then G contains an r-regular subgraph. APPENDIX
In this Section we prove a generalization of Olson’s theorem (Theorem 2.1), and apply it to graph theory. Let Z be the set of integers. For S c Z and m E Z, let card,(S) denote the number of distinct elements in S modulo m. The main result in this section is THEOREM A. 1. Let p be a prime and let d, > d, > . . . > d, be n positive integers. For 1 <j< n let Sj c Z be a set of integers containing 0. For 1 < i < m let a”’ = (a?‘,..., at’) be a vector with integer coordinates. If n m > x (pdj - card,,(Sj)), j=l
then there exists a subset 0 #I c { 1,2,..., m} and numbers Sj E S,i (1 <j < n) such that
1 {uji): i E I} = sj (modpdj)
for
l<j 5 (pj-1). j=l
See, for example, [7].
ACKNOWLEDGMENTS We would like to thank N. Linial and R. Meshulam for fruitful discussions.
REFERENCES 1. N. ALON, S. FRIEDLAND, AND G. KALAI, Every 4-regular graph plus an edge contains a 3-regular subgraph, J. Combin. Theory Ser. B 37 (1984), 92-93. 2. B. BOLLOB,&, “Extremal Graph Theory,” Academic Press, New York, 1978. 3. R. C. BAKER AND W. M. SCHMIDT, Diophantine problems in variables restricted to the values 0 and 1, J. Number Theory 12 (1980), 460-486. 4. J. A. BONDY AND U. S. R. MURTY, “Graph Theory with Applications,” Macmillan & Co. London. 1976.
REGULAR SUBGRAPHS
91
5. Z. I. BOREVICH AND I. R. SHAFAREVICH, “Number Theory,” Chap. 1, Academic Press, New York, 1966. 6. V. CHVATAL, H. FLEISCHNER,J. SHEEHAN, AND C. THOMASSEN, Three regular subgraphs of four regular graphs, J. Graph Theory 3 (1979), 371-386. 7. G. T. DIDERRICH AND H. B. MANN, Combinatorial problems in finite abelian groups, in “A Survey of Combinatorial Theory” (J. N. Srivastava et al., Eds.), pp. 95-100, NorthHolland, Amsterdam, 1973. 8. M. KANO AND A. SAITO, [a, b] factors of graphs, Discrete M&h. 47 (1983), 113-l 16. 9. J. E. OLSON, A combinatorial problem on finite abelian groups, J. Number Theory 1 (1969), 8-10. 10. J. E. OLSON, A combinatorial problem on finite abelian groups, II, J. Number Theory 1 (1969), 195-199. 11. V. A. TA~KINOV, Regular subgraphs of regular graphs, Soviet Muth. Dokl. 26 (1982), 37-38.
12. C. THOMASSEN, A remark on the factor theorems of Lovisz and Tutte, J. Graph Theory (1981), 441442. 13. W. T. TUNE, The subgraph problem, Ann. Discrete Math. 3 (1978), 289-295.
5
OF COMBINATORIAL
Regular
THEORY,
Series B 37, 79-91 (1984)
Subgraphs
of Almost N.
Department
Graphs
ALON*
of Mathematics, Massachusetts Institute Cambridge, Massachusetts
S. Institute
Regular
of Technology,
FRIEDLAND
of Mathematics, Hebrew University Jerusalem, Israel
of Jerusalem,
AND
G. Department
KALAI*
of Mathematics, Massachusetts Institute Cambridge, Massachusetts Communicated
by the Managing
of Technology,
Editors
Received July 25, 1983
Suppose every vertex of a graph G has degree k or k + 1 and at least one vertex has degree k + 1. It is shown that if k > 2q - 2 and q is a prime power then G contains a q-regular subgraph (and hence an r-regular subgraph for all r < q. r = q (mod 2)). It is also proved that every simple graph with maximal degree A > 2q - 2 and average degree d > ((2q - 2)/(2q - l))(A + 1), where q is a prime power, contains a q-regular subgraph (and hence an r-regular subgraph for all r < q, r = q (mod 2)). These results follow from Chevalley’s and Olson’s theorems on congruences. 0 1984 Academic Press, Inc.
1.
INTRODUCTION
In this paper we use the theorems of Chevalley [5] and Olson [9, lo] (and some extensions) on congruences, to prove the existence of regular subgraphs of certain graphs. All graphs considered are finite, undirected, and contain no loops, unless otherwise stated. Note that we allow multiple edges. A simple graph is a graph without multiple edges. * The contribution of these authors was supported in part by the Weizmann Fellowship for Scientific Research.
79 00958956/84 582b/37/1-6
$3.00
Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved.
80
ALON,
FRIEDLAND,
AND
KALAI
A graph H is q-divisible if q divides the degree of every vertex of H. Detinef(n, q) to be the maximal number of edges of a graph G on n vertices, that contains no nonempty q-divisible subgraph. In Section 3 we prove that
provided q is an odd prime power. If q is a power of 2, then
We further show that in both inequalities equality holds for all n > 3 and that a graph on n > 3 vertices and e >f(n, q) edges contains at least 2e-fh7) - 1 nonempty q-divisible subgraphs. Note that for q = 2 this is just the well-known fact that the dimension of the cycle space of G is at least e-n+ 1. For k < s a graph G is of type (k, s) if the degree d(v) of every vertex of it satisfies k < d(v) < s and G is not k-regular. In Section 4 we show that if q is a prime power, q > r, q = r (mod 2) and k > 2q - 2 then every graph G of type (k, k + 1) contains an r-regular subgraph. In particular: Every 4-regular graph plus one edge contains a 3-regular subgraph. (1.1) This result is closely related to a well-known conjecture of Berge and Sauer (see, e.g., [4, p. 2461) that asserts that every 4regular simple graph has a 3-regular subgraph. Some positive results about this conjecture can be found in [6], and in [ 111 Taikinov announced that he verified it. However, the Berge-Sauer conjecture is false for graphs with parallel edges; (every graph obtained from an odd cycle by replacing every edge by two parallel edges forms a counterexample). Therefore, the “plus one edge” cannot be omitted in (1.1). A short derivation of (I. 1) from Chevalley’s theorem appears separately in [ 11. In Section 4 we also show that if q is a prime-power, q > r and q = r (mod 2) then every simple graph G with maximal degree d > 2q - 2 and average degree d > ((29 - 2)/(2q - l))(d + 1) contains an r-regular subgraph. In particular, every simple graph with maximal degree d > 4 and average degree d > $(d + 1) contains a 3-regular subgraph. This result may help in solving a long standing problem of Erdiis and Sauer (see, e.g., [2, p. 399, problem 201). They asked for an estimation of the maximal number of edges of a simple graph on n vertices that contains no 3-regular subgraph. Our paper is organized as follows. In Section 2 we describe the algebraic tools: we quote the theorem of Olson, show how it is related to the classical theorem of Chevalley, and obtain a simple corollary. In the Appendix we prove an extension of Olson’s theorem and apply it to graph theory. Our proof is different from Olson’s proof and is somewhat similar to the proof of
81
REGULAR SUBGRAPHS
Chevalley’s theorem given in [5]. In Section 3 we derive the results on q-divisible subgraphs and in Section 4 we combine these with known results of Petersen, TaOkinov, Thomassen, Tutte, and Vizing to obtain our results on regular subgraphs.
2. THE ALGEBRAIC
TOOLS
Our main algebraic tool in this paper is the following result of Olson [9]. Let p be a prime and suppose d, > d, > . . . > THEOREM 2.1 (Olson). d, > 0. For 1 < i < m let a”’ = (a?‘, a:“,..., a’,“) be a vector with integer coordinates. If m > k (pdj-
l)
j=l
then there exists a subset 0 # Z c ( 1,2,..., m) such that
c {a;“: i E I} E 0 (modpdi),
j = l,..., n.
(2.1)
It is worth noting that for d, = d, = ..a = d, = 1 it is possible to derive this result from the classical theorem of Chevalley (see, e.g., [5]). Indeed consider the following system of polynomial equations 1$, aji’xf-’
= 0 (modp),
j=
I,..., n.
Clearly xi = 0 is a solution. Since the left-hand side of each equation is of degree p - 1 at most, the Chevalley’s theorem ensures a nontrivial solution if m > (p - 1) it. As xp-’ = 1 (mod p) for x f: 0 (modp), the existence of a nontrivial solution implies the assertion of Theorem 2.1 in this case. In the Appendix we prove a generalization of Olson’s theorem. Our proof is different from Olson’s proof and is somewhat similar to the proof of Chevalley’s theorem given in [5]. COROLLARY
2.2.
Suppose d, 2 d, > ... > d, > 1 and let p and a”’
(1 < i( m) be as in Theorem 2.1. If i j=l
a;!) G 0 (modp)
for
i = 1, 2,..., m
82
ALON,
FRIEDLAND,
AND
KALAI
and n-1 m
>p4-I
-
l+
x
(pdj-1)
j=l
then the conclusion of Theorem 2.1 holds. ProoJ For every 1 < i < m, define a vector b”’ = (bli),..., bc’) as follows: bji) = asi’ for 1 <j < n - 1 and bt’ = l/p CJ= I ali’. Applying Theorem 2.1 to the vectors b”), we conclude that there exists a subset 0 # Z c ( l,..., m) such that C {aj”: i E I} = c
{bji): i E I) s 0 (modpdj)
for 1 <j d, > ea. > d, , we conclude i E I} E 0 (modpdj) for all 1 <j < n. B
It is worth noting that both Theorem 2.1 and Corollary 2.2 are best possible. Indeed, let e”’ be the standard vector (Sj, ,..., Sj,), j = l,..., It. A set of pdj - 1 copies of e”’ for 1 <j < n shows that Theorem 2.1 is best possible. A set of pdj - 1 copies of e(j) - e’“’ for 1 Q j < n - 1 plus p d,-1 - 1 copies of pe W) shows that Corollary 2.2 is best possible. Theorem 2.1 and Corollary 2.2 can be used to prove the existence of one 0 # z c { 1, 2,..., m} that satisfies (2.1). Combining them with the following proposition of Olson [lo] we conclude that if m is large enough there are many such Is. PROPOSITION 2.3 (Olson). Let H be an abelian group and suppose that for every h I ,..., h,, , E H there exists a subset 0 # Z c { l,..., g + 1) such that
c (hi: i E I} = 0.
(2.2)
Zfh , ,..., h,,, E H and I > 1, then there exist at least 2’ - 1 distinct subsets Z, 0fzc {l,..., g + I} that satisfy (2.2). 3.
q-DIVISIBLE
SUBGRAPHS
Our main task in this section is to estimate the function f(n, q) defined in Section 1. Recall that f(n, q) is the maximum number of edges of a graph G
83
REGULARSUBGRAPHS
on n vertices that contains no nontrivial q-divisible subgraph. Clearly f(n, 1) = 0 and f(n, 2) = n - 1. The following theorem is an easy consequence of Theorem 2.1 and Corollary 2.2. THEOREM
3.1.
ft%4) 3. Define g(n,k)=(k-
1)-n
=(k-l)*n-(k/2)
if
k is odd,
if
k is even.
For an odd integer k > 1 let G,(k) denote the Shannon triangle obtained from a triangle by replacing each edge by k - 1 parallel edges. Similarly, for even k, let G,(k) be the graph obtained from a triangle by replacing two edges by k - 1 parallel edges each and the third edge by (k/2) - 1 parallel edges. For n > 3 let G, = (V,, E,J be a graph obtained from G,(k) by adding to it n - 3 new vertices and joining each by k - 1 edges to vertices of G,. Clearly 1V,l = n and JE,I = g(n, k). One can easily check that G, contains no nontrivial k-divisible subgraph. Combining this with Theorem 3.1 we obtain PROPOSITION
(ii)
3.2.
(i)
For n > 3 and every k, f(n, k) > g(n, k).
If q is a prime power then f(n, q) = g(n, q).
There is some interest in considering separately the case of simple graphs. Thus we definef,(n, q) as the maximal number of edges of a simple graph G on n vertices that contains no nontrivial q-divisible subgraphs. Clearly f,(n, q) q* - 1 equality holds.
84
ALON,
PROPOSITION 3.3.
(ii)
FRIEDLAND,
AND
KALAI
(i)
For every odd integer k > 1 and every n > k2 - 1
f,h
k) > g(n, k) = (k - 1) . n.
If q is an odd prime power
and n > q2 - 1 then
f,(n,q)=g(n,q)=(q-l).n. Proof. Part (ii) follows immediately from (i) and Theorem 3.1. To prove part (i) we construct a suitable example. Let tG denote the disjoint union of t copies of the graph G. Let G + H denote the join of the graphs G and H, i.e., the graph obtained from their disjoint union by joining each vertex of G to each vertex of H. Let Ek-, be the graph consisting of k - 1 isolated vertices, let K,,,- , denote the star with k - 1 edges and define G,,=JIZ~-~ + (k-
l)Kl,+l.
Suppose n > k2 - 1. Let G = (V, E) be a graph obtained from G, by adding to it n - (k2 - 1) new vertices and joining each of them to k - 1 vertices of G,. One can easily check that 1VI = n and 1E I= (k - 1) . n. In order to complete the proof we must show that G contains no nontrivial k-divisible subgraph. Clearly it is enough to show that G, contains no such graph. G, has vertices of three different types: let us call these of degree k(k - 1) vertices of type 1, these of degree 2k - 2- of type 2, and these of degree k- of type 3. Suppose G, has a nontrivial k-divisible subgraph H = (Y’, E’), where dH(v) > 0 for all v E V’. We claim that V’ contains all k - 1 vertices of type 1. Indeed, otherwise V’ contains no vertex of type 3 (since its degree in H is 1, then G contains at least 2’ - 1 nontrivial q-divisible subgraphs. 1
Note that since a 2-divisible
subgraph is just an Eulerian-subgraph,
for
q = 2 the last theorem is the well-known fact that the dimension of the cycle space of G is at least e - n + 1. Remark 3.6. Let G = (V, E) be a directed graph. For e E E and v E V put a:‘= + 1 (-1) if e goes out of (into) v and a:,@= 0 otherwise. If V = {V , ,***,v,,} define a’@ = (at’,..., a:). Since ,JJj”=, aI;’ = 0, one can apply Theorem 2.1 to the vectors (a!:‘,..., a~~‘-,) and show that if q is a prime power (even or odd) and JE1> (q - 1) e (n - 1), then G contains a subgraph H such that q 1d,+(v) - d;(v) for all u E V. This easily implies that every bipartite graph G with n vertices and more than (q - l)(n - 1) edges contains a nontrivial q-divisible subgraph.
We close this section with a conjecture. Conjecture 3.1.
For every n > 3 and every k f(n, k) < (k - 1) . n.
86
ALON,FRIEDLAND,AND
4. REGULAR
Recall the definition prove the following.
SUBGRAPHS OF ALMOST
KALAI REGULAR
GRAPHS
of a graph of type (k, s) given in Section 1. Here we
THEOREM 4.1. Let G be a graph of type (k, k + 1) and let r be an integer, If q is a prime power, q > r, q = r (mod 2), and k > 2q - 2 then G contains an r-regular subgraph. THEOREM 4.2. Let G be a graph of type (k, k + 2) and let r be an integer. If q is a prime power, q > r, q E r (mod 2), and k > 2q - 1 then G contains an r-regular subgraph.
Note that by Bertrand’s postulate (for every r there is a prime between r and 2r), Theorem 4.2 implies that if k > 4r then every graph of type (k, k + 2) contains an r-regular subgraph. In fact, the known improvements of Bertrand’s postulate (see, e.g., [2, p. xx]) enable one to show that Theorem 4.2 implies that for every E > 0 if r > r(s) is odd and k > (2 + E) r then every graph of type (k, k + 2) contains an r-regular subgraph. Shannon’s triangle obtained by replacing each edge of a triangle by r - 1 parallel edges shows that this is close to being best possible. For simple graphs we prove THEOREM 4.3. If q is aprimepower, q > r, and q s r (mod 2) then every simple graph G with maximal degree A > 2q - 2 and average degree d > ((29 - 2)/(2q - l))(A + 1) contains an r regular subgraph.
These theorems are proved by combining Theorem 3.1 with results of Petersen, TaBkinov, Thomassen, Tutte, and Vizing. In what follows we state these results and prove our theorems. LEMMA 4.4 (Thomassen [ 121, a somewhat weaker version was proved by Tutte [ 131.). Let G be a graph of type (k, k + 1) and suppose0 < r < k. Then G contains a spanning subgraph of type (r, r + 1). LEMMA 4.5 (Proved by Petersen [4, p. 751 for even k, and by Taikinov [ 111 for odd k.). If k > r, k = r (mod 2), then every k regular graph contains an r regular subgraph.
Proof of Theorem 4.1. By Lemma 4.4, G contains a (spanning) subgraph L=(V,E) of type (2q-2,29-l). Clearly ]E]>f.]v] (2q-2)= (q- 1). IV]. Th eref ore, by Theorem 3.1, L contains a nontrivial q-divisible subgraph H. However, for every v E V deg,(v) Q deg,(v) < 2q - 1 and thus H is q regular. The result now follows from Lemma 4.5. 1
REGULARSUBGRAPHS
87
For the proof of Theorem 4.2 we need LEMMA 4.6. Let G be a graph of type (k, k + 2) and suppose0 < I < k - 2. Then G contains a spanning subgraph of type (r, r + 2) with average degree strictly greater than r + 1.
Outline of Proof The proof is very similar to the proof of Thomassen [ 121 to Lemma 4.4. His argument easily shows that G contains a spanning subgraph L of type (r + 2, r + 4). The same argument shows that L contains a spanning subgraph H of type (r + 1, r t 3) with at least one vertex of degree r + 2, and that H has a spanning subgraph of type (r, r t 2) with more vertices of degree r t 2 than vertices of degree r. We omit the details. 1 Proof of Theorem 4.2. By Lemma 4.6, G contains a (spanning) subgraph L=(V,E)oftype(2q-3,2q-l)withIEl>(q-l)1VI.Thissubgraph,as is shown in the proof of Theorem 4.1, contains an r-regular subgraph. a
For the proof of Theorem 4.3 we need the following well-known result of Vizing (see, e.g., (2, pp. 230-2321). LEMMA 4.7 (Vizing). The edges of every simple graph with maximal degree A can be covered by A t 1 disjoint matchings.
Proof of Theorem 4.3. $ y L emma 4.7 the edges of G = (V, E) can be covered by A t 1 disjoint matchings. Let L = (V, E’) be the graph on V consisting of the edges of the 2q - 1 biggest matchings. Clearly
IE’l>~~E~=~.+d.lP’ (2q-1)(2q-2)(A+1)~Y~=(q-1)/YI. >2(A t 1) (2q - 1)
By Theorem 3.1, L contains a nontrivial q-divisible subgraph H, which is, as in the proof of Theorem 4.1, q-regular. The result follows from Lemma 4.5. I Remark 4.8. (a) We can slightly improve the constant (2q - l)/ (2q - 2) in Theorem 4.3 but this makes the proof somewhat more com-
plicated. (b) Remark 3.6 and Kiinig’s theorem (see, e.g., [4, p. 93, Theorem 6.11) enables us to obtain the following improvement of Theorem 4.3 for bipartite graphs: If q is a prime power, q > r then every bipartite graph G with maximal degree A and average degree d > ((2q - 2)/ (2q - 1)) A contains an r-regular subgraph.
88
ALON,FRIEDLAND,AND
KALAI
(c) Combining the result of [8] with Theorem 4.2 one can easily prove the following: Let G be a graph of type (k, k + s) and let r be an integer. If q is a prime power, q > r, q- r (mod 2), k> 2q - 1, and s/k < 2/(2q - l), then G contains an r-regular subgraph. APPENDIX
In this Section we prove a generalization of Olson’s theorem (Theorem 2.1), and apply it to graph theory. Let Z be the set of integers. For S c Z and m E Z, let card,(S) denote the number of distinct elements in S modulo m. The main result in this section is THEOREM A. 1. Let p be a prime and let d, > d, > . . . > d, be n positive integers. For 1 <j< n let Sj c Z be a set of integers containing 0. For 1 < i < m let a”’ = (a?‘,..., at’) be a vector with integer coordinates. If n m > x (pdj - card,,(Sj)), j=l
then there exists a subset 0 #I c { 1,2,..., m} and numbers Sj E S,i (1 <j < n) such that
1 {uji): i E I} = sj (modpdj)
for
l<j 5 (pj-1). j=l
See, for example, [7].
ACKNOWLEDGMENTS We would like to thank N. Linial and R. Meshulam for fruitful discussions.
REFERENCES 1. N. ALON, S. FRIEDLAND, AND G. KALAI, Every 4-regular graph plus an edge contains a 3-regular subgraph, J. Combin. Theory Ser. B 37 (1984), 92-93. 2. B. BOLLOB,&, “Extremal Graph Theory,” Academic Press, New York, 1978. 3. R. C. BAKER AND W. M. SCHMIDT, Diophantine problems in variables restricted to the values 0 and 1, J. Number Theory 12 (1980), 460-486. 4. J. A. BONDY AND U. S. R. MURTY, “Graph Theory with Applications,” Macmillan & Co. London. 1976.
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5. Z. I. BOREVICH AND I. R. SHAFAREVICH, “Number Theory,” Chap. 1, Academic Press, New York, 1966. 6. V. CHVATAL, H. FLEISCHNER,J. SHEEHAN, AND C. THOMASSEN, Three regular subgraphs of four regular graphs, J. Graph Theory 3 (1979), 371-386. 7. G. T. DIDERRICH AND H. B. MANN, Combinatorial problems in finite abelian groups, in “A Survey of Combinatorial Theory” (J. N. Srivastava et al., Eds.), pp. 95-100, NorthHolland, Amsterdam, 1973. 8. M. KANO AND A. SAITO, [a, b] factors of graphs, Discrete M&h. 47 (1983), 113-l 16. 9. J. E. OLSON, A combinatorial problem on finite abelian groups, J. Number Theory 1 (1969), 8-10. 10. J. E. OLSON, A combinatorial problem on finite abelian groups, II, J. Number Theory 1 (1969), 195-199. 11. V. A. TA~KINOV, Regular subgraphs of regular graphs, Soviet Muth. Dokl. 26 (1982), 37-38.
12. C. THOMASSEN, A remark on the factor theorems of Lovisz and Tutte, J. Graph Theory (1981), 441442. 13. W. T. TUNE, The subgraph problem, Ann. Discrete Math. 3 (1978), 289-295.
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